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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305616691

Chapter 17, Problem 1RCC

Textbook Problem

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(a) Write the general form of a second-order homogeneous linear differential equation with constant coefficients.

(b) Write the auxiliary equation.

(c) How do you use the roots of the auxiliary equation to solve the differential equation? Write the form of the solution for each of the three cases that can occur.

(a)

To determine

**To write:** The general form of a second-order homogeneous linear differential equation with constant coefficients.

The general form of a second-order homogeneous linear differential equation with constant coefficients is

**Formula used:**

Consider the second-order linear differential equation as follows.

Here,

If

Consider the value of

Substitute

Consider

Substitute

Thus, the general form of a second-order homogeneous linear differential equation with constant coefficients is

(b)

To determine

**To write**: The auxiliary equation.

The auxiliary equation of the second-order differential equation is

Modify equation (3) as follows.

In equation (4), function

Consider a exponential function for

Differentiate

Differentiate

Substitute

Since, the value of

Equation (6) is known as characteristic equation or auxiliary equation of the second-order differential equation

Thus, the auxiliary equation of the second-order differential equation is

(c)

To determine

**To explain:** The use of roots of the auxiliary equation to solve the differential equation and write the form of the solution for each of the three cases.

The use of roots of the auxiliary equation to solve the differential equation is explained.

The form of the solution for each of the three cases is written.

**Formula used:**

Consider the second-order differential equation as follows.

Write the expression for quadratic formula.

Modify equation (7) as follows.

Modify equation (8) as follows.

Thus, the use of roots of the auxiliary equation to solve the differential equation is explained.

Three different cases are obtained depending upon the term

**Case I:**

Consider the value of

In this case, the roots of auxiliary equation are real and distinct. So two linear independent solutions are occurs such as

Write the expression for general solution.

Here,

Substitute

**Case II:**

Consider the value of

In this case, the roots of auxiliary equation are real and equal.

Consider

Substitute

**Case III:**

Consider the value of

In this case, the roots of auxiliary equation are complex numbers.

Consider

Write the expression for general solution with complex roots.

Thus, the form of the solution for each of the three cases is written.

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